4. A new approach to the classification problem for multiplier algebras 61
4.9. Holomorphic maps on homogeneous varieties
In the last section, we saw that the maximal ideal space of an algebra of the type A(HI) or Mult(HI) contains a copy of the homogeneous variety Z0(I). We will see in the next section that under suitable conditions, algebra homomorphisms between our algebras in-duce holomorphic maps between the varieties. Thus, we will require some results about holomorphic maps on homogeneous varieties. The arguments presented in the first part of this section (up to Lemma 4.9.6) already appeared in the author’s Master’s thesis [41, Section 3.3].
4.9. Holomorphic maps on homogeneous varieties
Throughout this section, let I, J ( C[z1, . . . , zd] be radical homogeneous ideals. We say that a map F : Z0(I) → Cd0, where d0 ∈ N, is holomorphic if for every z ∈ Z0(I), there exists an open neighbourhood U of z and a holomorphic function G on U which agrees with F on U ∩ Z0(I).
We require the following variant of the maximum modulus principle.
Lemma 4.9.1. Let F : Z0(I) → Bd be a holomorphic map. If F is not constant, then F (Z0(I)) ⊂ Bd.
Proof. We may assume that {0} ( Z0(I). Suppose that there exists w ∈ Z0(I) such that
||F (w)|| = 1 and choose w ∈ Ze 0(I) satisfying w ∈ Dw. The ordinary maximum moduluse principle shows that the holomorphic function
D → D, t 7→ hF (tw), F (w)i,e
is the constant function 1. Consequently, F (tw) = F (w) for all t ∈ D, and in particulare F (0) = F (w) ∈ ∂Bd. Now, if z ∈ Z0(I) is arbitrary, another application of the maximum modulus principle shows that the function
D → D, t 7→ hF (tz), F (0)i,
is the constant function 1, hence F (z) = F (0). Thus, F is constant.
The next goal is to show that every biholomorphism between Z0(I) and Z0(J ) which fixes the origin is the restriction of an invertible linear map. This result is Theorem 7.4 in [24], where it was established by adjusting the proof of Cartan’s uniqueness theorem from [75, Theorem 2.1.3]. We provide a simpler proof, which only uses the Schwarz lemma from ordinary complex analysis. We begin with the following variant of the Schwarz lemma.
Lemma 4.9.2. Let d0 ∈ N and let F : Z0(I) → Bd0 be a holomorphic map such that F (0) = 0. Then ||F (z)|| ≤ ||z|| for all z ∈ Z0(I). If equality holds for some z ∈ Z0(I)\{0}, then there exists w0 ∈ ∂Bd0 such that
F t z
||z||
= tw0 (4.5)
for all t ∈ D. In particular, F maps the disc Cz ∩ Bd biholomorphically onto the disc CF (z) ∩ Bd0 in this case.
Proof. We may assume that {0} ( Z0(I). Let z ∈ Z0(I) \ {0}, suppose that F (z) 6= 0 and define w0 = F (z)/||F (z)||. By the classical Schwarz lemma, the function
f : D → D, t 7→D F
t z
||z||
, w0E
,
satisfies |f (t)| ≤ |t| for all t ∈ D. The first statement now follows by choosing t = ||z||.
If ||F (z)|| = ||z||, then f (||z||) = ||z||, thus f is the identity by the Schwarz lemma. Since
||F (t||z||z )|| ≤ |t| for all t ∈ D by the first part, Equation (4.5) holds. The last assertion is now obvious.
The desired result about biholomorphisms which fix the origin follows as an application of the last lemma.
Proposition 4.9.3 ([24, Theorem 7.4]). Let F : Z0(I) → Z0(J ) be a biholomorphism such that F (0) = 0. Then there exists an invertible linear map A on Cd which maps V (I) isometrically onto V (J ) such that A
Z0(I) = F .
Proof. We may again assume that {0} ( Z0(I). Let G be a holomorphic map which is defined on a neighbourhood U of 0 and which coincides with F on U ∩ Z0(I). Let A0 be the derivative of G at 0. Lemma4.9.2, applied to F and its inverse, shows that ||F (z)|| = ||z||
for all z ∈ Z0(I), so the second part of the same lemma applies. Taking the derivative with respect to t in Equation (4.5) for fixed z ∈ Z0(I) \ {0}, we see that w0 necessarily satisfies w0||z|| = A0z, hence
F (z) = ||z||w0 = A0z.
Thus, A0
Z0(I) = F , and A0 is isometric on Z0(I) since F is. Linearity of A0 implies that A0 maps V (I) isometrically onto V (J ).
Finally, the same argument, applied to F−1 in place of F , shows that there exists a linear map B0 on Cd such that B0
Z0(J )= F−1. From this, we deduce that A0 restricts to a linear isomorphism from the linear span of Z0(I) onto the linear span of Z0(J ). Thus, if we let A be an invertible extension of A0
Z0(I) to Cd, then A satisfies all the requirements of the proposition.
We also crucially require a result from [24], which, loosely speaking, allows us to repair biholomorphisms which do not fix the origin. This result is contained in the proof of Proposition 4.7 in [24]. The proof in [24] proceeds in two steps. In a first step, tools from algebraic geometry and knowledge about the structure of conformal automorphisms of Bd are used to reduce the statement about arbitrary homogeneous varieties to the case
4.9. Holomorphic maps on homogeneous varieties
of discs. The second step, which deals with the case of discs, is an argument from plane conformal geometry.
It turns out that the first step, namely the reduction to discs, also follows immediately from Lemma 4.9.2.
Lemma 4.9.4. Let F : Z0(I) → Z0(J ) be a biholomorphism with F (0) 6= 0. Let b = F (0) and let a = F−1(0). Then ||a|| = ||b|| and F maps the disc D1 = Ca ∩ Bd biholomorphically onto the disc D2 = Cb ∩ Bd.
Proof. Let
f : D → Z0(J ), t 7→ F t a
||a||
,
and let ϕ be an automorphism of D which maps 0 to ||a|| and vice versa. Then h = f ◦ ϕ satisfies the assumptions of Lemma4.9.2, hence
||b|| = ||h(||a||)|| ≤ ||a||.
By symmetry, ||a|| ≤ ||b||, so ||a|| = ||b||. It now follows from the second part of Lemma 4.9.2 that h maps D biholomorphically onto the disc D2. The result follows.
The second step is essentially the following lemma. For λ ∈ T, let Uλ denote the unitary map on Cd defined by
Uλ(z) = λz for z ∈ Cd.
Lemma 4.9.5 (Davidson-Ramsey-Shalit [24]). Let ϕ be a conformal automorphism of D.
The set
{(Uλ◦ ϕ−1◦ Uµ◦ ϕ)(0) : λ, µ ∈ T} ⊂ D is a closed disc around 0 which contains the point ϕ−1(0).
Proof. We repeat the relevant part of the proof of Theorem 7.4 in [24]. The assertion is trivial if ϕ fixes the origin, so we may assume that ϕ(0) 6= 0. Then
C = {(Uµ◦ ϕ)(0) : µ ∈ T}
is the circle around 0 with radius |ϕ(0)|. Since automorphisms of D map circles to circles, it follows that the set ϕ−1(C) is a circle which obviously passes through 0. Moreover, ϕ−1(0) is contained in the interior of the circle ϕ−1(C) as 0 is contained in the interior of C. Thus
{(Uλ◦ ϕ−1◦ Uµ◦ ϕ)(0) : λ, µ ∈ T} = {Uλ(ϕ−1(C)) : λ ∈ T}
is a closed disc around 0 which contains ϕ−1(0).
Observe that if I ( C[z1, . . . , zd] is a radical homogeneous ideal, then Uλ leaves Z0(I) and Z(I) invariant for each λ ∈ T. Combining Lemmata 4.9.4 and 4.9.5, we obtain the result from [24] which allows us to repair biholomorphisms which do not fix the origin.
Lemma 4.9.6 (Davidson-Ramsey-Shalit [24]). Let I, J ( C[z1, . . . , zd] be radical homo-geneous ideals and suppose that F : Z0(I) → Z0(J ) is a biholomorphism. Then there are λ, µ ∈ T such that the biholomorphism
F ◦ Uλ◦ F−1◦ Uµ◦ F : Z0(I) → Z0(J ) fixes the origin.
Proof. The assertion is trivial if F (0) = 0, so we may assume that F (0) 6= 0. It follows then from Lemma4.9.4that it suffices to consider the case where d = 1 and where Z0(I) = Z0(J ) = D, the unit disc. An application of Lemma 4.9.5 shows that there are λ, µ ∈ T such that
F−1(0) = (Uλ◦ F−1◦ Uµ◦ F )(0), hence F ◦ Uλ◦ F−1◦ Uµ◦ F fixes the origin.
We finish this section by giving another application of the crucial Lemma4.9.5of David-son, Ramsey and Shalit [24]. We will show that the group of unitaries is a maximal sub-group of Aut(Bd), the group of conformal automorphisms of Bd. Since the group Aut(Bd) is well studied, it is likely that this has been observed before. Nevertheless, even when d = 1, the only result in this direction that seems to be widely known is the fact that the group of unitaries is a maximal compact subgroup of Aut(Bd).
Recall that for a ∈ Bd, there exists an automorphism ϕa of Bd defined by ϕa(z) = a − Paz − saQaz
1 − hz, ai (z ∈ Bd),
where Pa is the orthogonal projection of Cd onto the subspace spanned by a, Qa= I − Pa and sa = (1 − |a|2)1/2. Then ϕa is an involution which interchanges 0 and a (see, for example, [75, Theorem 2.2.2]). Moreover, every ϕ ∈ Aut(Bd) is of the form ϕ = U ◦ ϕa, where U is unitary and a = ϕ−1(0) [75, Theorem 2.2.5]. We begin with a preliminary lemma.
Lemma 4.9.7. Let G ⊂ Aut(Bd) be a subsemigroup which contains all unitary maps and let O denote the orbit of 0 under G. Then the following assertions hold:
(a) G is a subgroup of Aut(Bd).
4.9. Holomorphic maps on homogeneous varieties
(b) A point a ∈ Bd belongs to O if and only if ϕa∈ G.
(c) G = Aut(Bd) if and only if O = Bd.
Proof. (a) If ϕ ∈ G, then ϕ = U ϕa for some unitary map U and a ∈ Bd. Then ϕa ∈ G.
Since ϕa is an involution, it follows that ϕ−1 = ϕaU−1 ∈ G. Hence, G is a group.
(b) For the proof of the non-trivial implication, suppose that a ∈ O and let ϕ ∈ G with a = ϕ(0). Then ϕ−1 ∈ G by part (a) and (ϕ−1)−1(0) = a, hence
ϕ−1 = U ◦ ϕa
for some unitary map U . Since U ∈ G, it follows that ϕa∈ G, as asserted.
(c) This follows immediately from (b) and the description of the automorphisms of Bd
in terms of unitary maps and the involutions ϕa.
We now show that the group of rotations is a maximal subgroup of the group Aut(D).
We will then deduce the higher-dimensional analogue from this result.
Lemma 4.9.8. The group of rotations is a maximal subgroup of the group Aut(D).
Proof. Let G be a subgroup of Aut(D) which properly contains the group of rotations. Let O be the orbit of 0 under G. We wish to show that O = D, which is equivalent to the assertion by part (c) of Lemma 4.9.7.
We first claim that DO ⊂ O. To this end, let a ∈ O. Then ϕa ∈ G by part (b) of Lemma 4.9.7. An application of Lemma 4.9.5 now shows that O contains the closed disc of radius |a| around 0, which proves the claim.
We finish the proof by showing that O contains points of modulus arbitrarily close to 1. Since G contains a non-rotation automorphism, O 6= {0}. Clearly, O is rotationally invariant, hence there exists r > 0 such that r ∈ O and therefore ϕr ∈ G by part (b) of Lemma4.9.7. Consider the hyperbolic automorphism f defined by
f (z) = ϕr(−z) = r + z 1 + rz
for z ∈ D. Then f ∈ G. Moreover, it is well known and easy to see that
n→∞lim fn(0) = 1,
where fn denotes the n-fold iteration of f . Thus, the proof is complete.
We are now ready to prove a multivariate analogue of the last lemma.
Proposition 4.9.9. The group of unitary maps on Cd is a maximal subsemigroup of Aut(Bd).
Proof. Suppose that G is a subsemigroup of Aut(Bd) which properly contains the group of unitary maps. Then G is a subgroup by part (a) of Lemma 4.9.7. Let O denote the orbit of 0 under G. Our goal is to show that O = Bd (see part (c) of Lemma 4.9.7). Since G contains all unitaries, it suffices to show that De1 ⊂ O, where e1 denotes the first standard basis vector of Cd.
To this end, let
H = {ϕ ∈ G : ϕ(De1) = De1}.
Identifying De1 with D we obtain a subgroup H = {ϕe
D: ϕ ∈ H}
of Aut(D). Clearly, eH contains all rotations Uλ for λ ∈ T. Since G contains a non-unitary automorphism, {0} 6= O. Moreover, O is invariant under unitary maps, hence there exists r > 0 such that re1 ∈ O and thus ϕre1 ∈ G by part (b) of Lemma 4.9.7. Observe that ϕre1 ∈ H, so eH contains the non-rotation automorphism ϕr. It now follows from Lemma 4.9.8 that eH = Aut(D). Since Aut(D) acts transitively on D, the definition of eH implies that De1 ⊂ O, which completes the proof.
There is an immediate consequence for collections of functions on Bdwhich are unitarily invariant.
Corollary 4.9.10. Let S 6= ∅ be a collection of functions on Bd and define G = {ϕ ∈ Aut(Bd) : f ◦ ϕ ∈ S for all f ∈ S}.
Assume that G contains U , the group of unitary maps on Cd. Then either G = U or G = Aut(Bd).
Proof. It is clear that G is a subsemigroup of Aut(Bd), so the result follows from Proposition 4.9.9.
The last result applies in particular to reproducing kernel Hilbert spaces H on Bd with a kernel of the form
K(z, w) =
∞
X
n=0
anhz, win (z, w ∈ Bd).
4.10. Existence of graded isomorphisms
In this case, by the closed graph theorem, G is also the set of all automorphisms of Bd
which induce a bounded composition operator on H. Moreover, G contains all unitaries.
Thus, the last result says that such a space H is either invariant under all automorphisms of Bd, or under unitaries only.
4.10. Existence of graded isomorphisms
The question of when two algebras of the type Mult(HI) are isomorphic is more difficult than the question about equality of multiplier algebras studied in Section 4.6. The chief reason is that isomorphisms do not necessarily respect the grading. Thus, our goal is to establish the existence of graded isomorphisms. As in [24], this will follow from an application of Lemma4.9.6.
Throughout this section, let H and K be unitarily invariant complete NP-spaces on Bd
or on Bd and let I, J ( C[z1, . . . , zd] be radical homogeneous ideals. We allow the case where H is a space on Bd, and K is a space on Bd, or vice versa. We will consider the multiplier algebras Mult(HI) and Mult(KJ), as well as their norm closed versions A(HI) and A(KJ). To cover both cases, we first study homomorphisms from A(HI) into Mult(KJ).
We identify the maximal ideal space of A(HI) with Z(I) by Lemma 4.8.1. Similarly, we identify Z0(J ) with a subset of the maximal ideal space of Mult(KJ) via point evaluations.
The following lemma should be compared to Proposition 7.1 and Lemma 11.5 in [24].
Lemma 4.10.1. Let H and K as well as I, J ( C[z1, . . . , zd] be as above.
(a) If Φ : A(HI) → Mult(KJ) is an injective unital homomorphism, then Φ∗ maps Z0(J ) holomorphically into Z0(I).
(b) If Φ : Mult(HI) → Mult(KJ) is an injective unital homomorphism and weak-∗-weak-∗
continuous, then Φ∗ maps Z0(J ) holomorphically into Z0(I).
(c) If Φ : Mult(HI) → Mult(KJ) is an injective unital homomorphism, and if H is tame, then Φ∗ maps Z0(J ) holomorphically into Z0(I), and Φ is weak-∗-weak-∗ continuous.
Proof. (a) Clearly, Φ∗ maps Z0(J ) into Z(I), and the j-th coordinate of Φ∗ is given by Φ(zj) ∈ A(HI), hence F = Φ∗
Z0(J )is holomorphic. Lemma4.9.1shows that the range of F contains points in ∂Bd only if F is constant. In this case, Φ(zj) = λj, where (λ1, . . . , λd) ∈
∂Bd. Since Φ is unital and injective, it follows that λj = zj on Z0(I), which is absurd.
Thus, the range of F is contained in Z0(I).
(b) By definition of the map π : M(Mult(HI)) → Z(I), part (a) implies that π ◦ Φ∗ is holomorphic and maps Z0(J ) into Z0(I). Since Φ is weak-∗-weak-∗ continuous, Φ∗(Z0(J )) consists of point evaluations by Lemma 4.5.7, so the assertion follows.
(c) Again by part (a), π ◦ Φ∗ is holomorphic and maps Z0(J ) into Z0(I). Since H is tame, we conclude that Φ∗ maps Z0(J ) into the set (of point evaluations at points in) Z0(I) (see Remark 4.8.4). If K is a space on Bd, Lemma 4.5.7 therefore implies that Φ is weak-∗-weak-∗ continuous. Now, assume that K is a space on Bd. If H is a space on Bd as well, then Φ∗(Z(J )) ⊂ Z(I) by continuity of Φ∗, thus Φ is again weak-∗-weak-∗ continuous by Lemma 4.5.7.
It remains to consider the case where H is a space on Bd and K is space on Bd. We claim that Φ∗(Z0(J )) is contained in a ball of radius r < 1. This will finish the proof, as Φ∗(Z(J )) ⊂ rZ(I) ⊂ Z0(I) by continuity, so once again, the assertion follows from Lemma 4.5.7. Suppose that Φ∗(Z0(J )) contains a sequence (Φ∗(λn)) with ||Φ∗(λn)|| → 1.
By passing to a subsequence, we may assume that (λn) converges to a point λ ∈ Z(J ).
Lemma 4.8.2 shows that there is a multiplier ϕ ∈ Mult(HI) such that (ϕ(Φ∗(λn))) does not converge. However,
ϕ(Φ∗(λn)) = (Φ(ϕ))(λn),
and Φ(ϕ) ∈ Mult(KJ) is a continuous function on Z(J ). This is a contradiction, and the proof is complete.
For isomorphisms, we obtain the following consequence.
Corollary 4.10.2. Let H and K as well as I, J ( C[z1, . . . , zd] be as above.
(a) If Φ : A(HI) → A(HJ) is an isomorphism, then Φ∗ maps Z0(J ) biholomorphically onto Z0(I).
(b) Let Φ : Mult(HI) → Mult(HJ) be an isomorphism, and assume that H is tame or that Φ is weak-∗-weak-∗ continuous. Then Φ∗ maps Z0(J ) biholomorphically onto Z0(I), and Φ is a weak-∗-weak-∗ homeomorphism.
Proof. (a) immediately follows from part (a) of the preceding lemma.
(b) By part (c) of the last lemma, Φ is weak-∗-weak-∗ continuous in both cases. Since it is also a homeomorphism in the norm topologies, the Krein-Smulian theorem combined with weak-∗ compactness of the unit balls shows that Φ−1 is weak-∗-weak-∗ continuous as well (see, for example, the argument at the end of the proof of Theorem3.2.5). Thus, part (b) of the last lemma also applies to Φ−1, so that Φ∗ is a biholomorphism between Z0(J ) and Z0(I).
For n ∈ N, let (HI)n denote the space of all homogeneous elements of HI of degree n.
Recall that (HI)n ⊂ A(HI) for all n ∈ N. We say that a homomorphism Φ : A(HI) → Mult(KJ) is graded if
Φ((HI)n) ⊂ (KJ)n
4.10. Existence of graded isomorphisms
for all n ∈ N. Graded isomorphisms admit a particularly simple description in terms of their adjoint.
Lemma 4.10.3. Let H and K as well as I, J ( C[z1, . . . , zd] be as above, and suppose that Φ : A(HI) → A(KJ) is an isomorphism (respectively that Φ : Mult(HI) → Mult(KJ) is a weak-∗-weak-∗ continuous isomorphism). Then the following are equivalent:
(i) Φ is graded.
(ii) Φ∗(0) = 0.
(iii) There exists an invertible linear map A on Cd which maps V (J ) isometrically onto V (I) such that Φ is given by composition with A, that is,
Φ(ϕ) = ϕ ◦ A for all ϕ ∈ A(HI) (respectively ϕ ∈ Mult(HI)).
Proof. (iii) ⇒ (i) is obvious.
(i) ⇒ (ii) Let λ = Φ∗(0) ∈ Z(I). If λ 6= 0, then there is a homogeneous element ϕ ∈ A(HI) of degree 1 such that ϕ(λ) 6= 0. Corollary 4.10.2 implies that Φ∗(0) ∈ Z0(I), hence
Φ(ϕ)(0) = ϕ(Φ∗(0)) = ϕ(λ) 6= 0.
In particular, Φ(ϕ) is not homogeneous of degree 1, hence Φ is not graded.
(ii) ⇒ (iii) By Corollary 4.10.2, Φ∗ maps Z0(J ) biholomorphically onto Z0(I). Since Φ∗(0) = 0, Proposition 4.9.3therefore yields an invertible linear map A which maps V (J ) isometrically onto V (I) such that Φ∗ coincides with A on Z0(J ). It follows that
Φ(ϕ) = ϕ ◦ A
on Z0(J ) for all ϕ ∈ A(HI) (respectively ϕ ∈ Mult(HI)). Moreover, if Φ is a map from A(HI) onto A(KJ), then this identity holds on Z(J ) by continuity.
Assume now that Φ is a map from Mult(HI) onto Mult(KJ). If K is a space on Bd, we are done. If K and H are spaces on Bd, then Φ(ϕ) = ϕ ◦ A again holds on Z(J ) by continuity. We finish the proof by showing that the remaining case where K is a space on Bd, H is a space on Bd and V (J ) 6= {0} does not occur. Indeed, in this case, V (I) 6= {0}
and Φ∗ would map Z(J ) onto a necessarily compact subset of Z0(I) by Lemma4.5.7. This contradicts the fact that Φ∗ maps Z0(J ) onto Z0(I).
We mention that in the case where H = K = Hd2, the Drury-Arveson space, isomorphisms as above are called vacuum-preserving in [24].
The desired consequence about the existence of graded isomorphisms is the following result.
Proposition 4.10.4. Let H and K as well as I, J ( C[z1, . . . , zd] be as above.
(a) If A(HI) and A(KJ) are algebraically (respectively isometrically) isomorphic, then there exists a graded algebraic (respectively isometric) isomorphism from A(HI) onto A(KJ).
(b) If Mult(HI) and Mult(KJ) are algebraically (respectively isometrically) isomorphic via a weak-∗ continuous isomorphism, then there exists a graded weak-∗-weak-∗ continuous algebraic (respectively isometric) isomorphism from Mult(HI) onto Mult(KJ).
Proof. By Lemma4.10.3, it suffices to show in each case that there exists an isomorphism whose adjoint fixes the origin. We will achieve this by applying Corollary 4.10.2 and Lemma 4.9.6. To this end, observe that for λ ∈ T, the unitary map Uλ on Cd given by multiplication with λ induces a unitary composition operator CUλ on HI. If ϕ ∈ Mult(HI), then
CUλMϕCU∗λ = Mϕ◦Uλ,
hence ΦIλ(Mϕ) = CUλMϕCU∗λdefines an isometric, weak-∗-weak-∗ continuous automorphism of Mult(HI) which maps A(HI) onto A(HI). Clearly, the adjoint of this automorphism, restricted to Z0(I), is given by multiplication with Uλ. The same result holds for KJ in place of HI.
Suppose now that Φ is an isomorphism between A(HI) and A(KJ) (respectively a
weak-∗-weak-∗ continuous isomorphism between Mult(HI) and Mult(KJ)). By Corollary 4.10.2, the adjoint Φ∗ maps Z0(J ) biholomorphically onto Z0(I). From Lemma 4.9.6, we infer that there exist λ, µ ∈ T such that the map
Φ∗◦ Uλ◦ (Φ∗)−1◦ Uµ◦ Φ∗ fixes the origin. This map is the adjoint of
Φ ◦ ΦIµ◦ Φ−1◦ ΦJλ ◦ Φ,
which is an isomorphism between A(HI) and A(KJ) (respectively a weak-∗-weak-∗ con-tinuous isomorphism between Mult(HI) and Mult(KJ)). Moreover, it is isometric if Φ is isometric, which finishes the proof.
4.11. Isomorphism results
4.11. Isomorphism results
We are now ready to establish the main results about isomorphism of multiplier algebras of spaces of the type HI. We will usually make an assumption which guarantees that the Hilbert function spaces have dimension at least 2. In projective algebraic geometry, the maximal ideal of C[z1, . . . , zd] which is generated by the coordinate functions z1, . . . , zd is called the irrelevant ideal (see [90, Chapter VII]). This is because the vanishing locus of this ideal in Cd is just the origin, hence the projective vanishing locus in Pd−1(C) is empty.
We will say that a radical homogeneous ideal of C[z1, . . . , zd] is relevant if it is proper and not equal to the irrelevant ideal. By the projective Nullstellensatz, the projective vanishing locus of every such ideal I is not empty, thus Z0(I) ⊂ Cd always contains a disc.
Proposition 4.11.1. Let H and K be unitarily invariant complete NP-spaces, and let I and J be relevant radical homogeneous ideals in C[z1, . . . , zd]. Let Φ : A(HI) → A(KJ) be a graded algebraic isomorphism (respectively Φ : Mult(HI) → Mult(KJ) a graded weak-∗-weak-∗ continuous isomorphism).
Then H = K as vector spaces, and there exists an invertible linear map A which maps V (J ) isometrically onto V (I) such that Φ is given by composition with A. Moreover, A induces a bounded invertible composition operator
CA : HI → KJ, f 7→ f ◦ A, such that
Φ(Mϕ) = CAMϕ(CA)−1
for all ϕ ∈ A(HI) (respectively ϕ ∈ Mult(HI)). In particular, Φ is given by a similarity.
Proof. By Lemma 4.10.3, there exists an invertible linear map A which maps V (J ) iso-metrically onto V (I) and such that Φ is given by composition with A. Since all Banach algebras under consideration are semi-simple, Φ and its inverse are (norm) continuous (see [17, Proposition 4.2]). Thus, if f ∈ HI is homogeneous, then Proposition 4.6.4 shows that
||f ◦ A||KJ = ||f ◦ A||Mult(KJ)≤ ||Φ|| ||f ||Mult(HI)= ||Φ|| ||f ||HI, so there exists a bounded operator CA : HI → KJ such that
CAf = f ◦ A
holds for every polynomial f , and hence for all f ∈ HI. Consideration of Φ−1 shows that CA is invertible. Moreover, for ϕ ∈ Mult(HI) and f ∈ KJ, we have
CAMϕ(CA)−1f = (ϕ ◦ A)f,
hence Φ is given by conjugation with CA.
We finish the proof by showing that H and K coincide as vector spaces. To this end, let KH(z, w) =
∞
X
n=0
anhz, win
and
KK(z, w) =
∞
X
n=0
a0nhz, win
denote the reproducing kernels of H and K, respectively. Since I and J are radical, Lemma 4.7.4implies that the restriction maps RI : H I → HI and RJ : K J → KJ are unitary.
Let
TA= R−1I (CA)∗RJ ∈ B(K J, H I).
Then TA is bounded and invertible, and Lemma 4.7.5combined with Lemma 4.7.4 implies that
TAKK(·, w) = KH(·, Aw)
for all w ∈ Z0(J ). Using the homogeneity of J , it is easy to deduce from KK(·, w) ∈ K J for w ∈ Z0(J ) that h·, win∈ K J for all w ∈ Z0(J ) and all n ∈ N. Similarly, h·, zin ∈ H I for all z ∈ Z0(I) and all n ∈ N. Moreover, CA and hence TA respects the degree of homogeneous polynomials. Consequently,
TAa0nh·, win = anh·, Awin (4.6) for all n ∈ N and all w ∈ V (J). Using part (d) of Remark 4.7.1 and the fact that
||Aw|| = ||w||, we see that
||a0nh·, win||2KJ = a0n||w||2n and that
||anh·, Awin||2H
I = an||w||2n. Since J is relevant, V (J ) contains a non-zero vector w, hence
||(CA∗)−1||2 ≤ an
a0n ≤ ||CA∗||2
for all n ∈ N, from which it immediately follows that H = K as vector spaces (see part (d) of Remark 4.7.1).
Using the same methods as in the last proof, we obtain a version of Proposition 4.11.1 for isometric isomorphisms.
4.11. Isomorphism results
Proposition 4.11.2. Let H and K be unitarily invariant complete NP-spaces, and let I and J be relevant radical homogeneous ideals in C[z1, . . . , zd]. Let Φ : A(HI) → A(KJ) be a graded isometric isomorphism (respectively Φ : Mult(HI) → Mult(KJ) a graded weak-∗-weak-∗ continuous isometric isomorphism).
Then H = K as Hilbert spaces, and there exists a unitary map U which maps V (J ) onto V (I) such that Φ is given by composition with U . Moreover, U induces a unitary composition operator
CU : HI → KJ, f 7→ f ◦ U, such that
Φ(Mϕ) = CUMϕ(CU)−1
for all ϕ ∈ A(HI) (respectively ϕ ∈ Mult(HI)). In particular, Φ is unitarily implemented.
Proof. Proposition 4.11.1 and its proof show that there exists an invertible linear map U which maps V (J ) isometrically onto V (I) such that U induces a unitary composition operator
CU : KJ → HI, f 7→ f ◦ U,
and such that Φ is given by conjugation with CU. Since CU is a unitary operator, the last part of the proof of Proposition 4.11.1shows that an= a0n for all n ∈ N in the notation of the proof, and hence H = K as Hilbert spaces.
Finally, setting n = 1 in Equation (4.6), we see that TUh·, wi = h·, U wi
for all w ∈ V (J ), and hence for all w in the linear span of V (J ). Since TU is a unitary operator, part (d) of Remark4.7.1 implies that U is isometric on the linear span of V (J ).
Hence, U is a unitary map from the linear span of V (J ) onto the linear span of V (I).
Changing U on the orthogonal complement of span(V (J )) if necessary, we can therefore achieve that U is a unitary map on Cd.
The last result, combined with Proposition4.10.4, provides a necessary condition for the existence of an isometric isomorphism between two algebras of the form A(HI), namely condition (iii) in the next theorem. This condition turns out to be sufficient as well. We thus obtain our main result regarding the isometric isomorphism problem. It generalizes [24, Theorem 8.2]. For a bounded invertible operator S between two Hilbert spaces H and K, let
Ad(S) : B(H) → B(K), T 7→ ST S−1, be the induced isomorphism between B(H) and B(K).
Theorem 4.11.3. Let H and K be unitarily invariant complete NP-spaces, and let I and J be relevant radical homogeneous ideals in C[z1, . . . , zd]. Then the following are equivalent:
(i) A(HI) and A(KJ) are isometrically isomorphic.
(ii) Mult(HI) and Mult(KJ) are isometrically isomorphic via a weak-∗-weak-∗ continuous isomorphism.
(iii) H = K as Hilbert spaces and there is a unitary map U on Cd which maps V (J ) onto V (I).
If H or K is tame, then this is equivalent to
(iv) Mult(HI) and Mult(KJ) are isometrically isomorphic.
If U is a unitary map on Cd as in (iii), then U induces a unitary composition operator CU : HI → KJ, f 7→ f ◦ U,
and Ad(CU) maps A(HI) onto A(KJ) and Mult(HI) onto Mult(KJ).
Proof. It follows from Proposition4.10.4and Proposition4.11.2that (i) or (ii) implies (iii).
Moreover, if one of the spaces is tame, then Corollary 4.10.2 (b) shows the equivalence of (ii) and (iv).
Conversely, suppose that (iii) holds. Since H = K is unitarily invariant, U induces a unitary composition operator bCU ∈ B(H). If K denotes the reproducing kernel of H, then
( bCU)∗K(·, w) = K(·, U w)
for all w ∈ Z0(J ) (or w ∈ Z(J ) if H is a space on Bd). Since H I and H J are spanned by kernel functions (see Lemma 4.7.4), the implication (ii) ⇒ (i) in Lemma 4.7.5 shows that U induces a unitary composition operator CU : HI → HJ.
for all w ∈ Z0(J ) (or w ∈ Z(J ) if H is a space on Bd). Since H I and H J are spanned by kernel functions (see Lemma 4.7.4), the implication (ii) ⇒ (i) in Lemma 4.7.5 shows that U induces a unitary composition operator CU : HI → HJ.